Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
Multicriteria HR Allocation Based on Hesitant
Fuzzy Sets and Possibilistic Programming
Zoran Ciric, Dragan Stojic, Otilija Sedlak
Faculty of Economics Subotica, Segedinski put 9-11, 24000 Subotica, Serbia
e-mail: zoran.ciric@ef.uns.ac.rs, stojicd@ef.uns.ac.rs, otilijas@ef.uns.ac.rs
Abstract: We focus in this paper on posing and solving a problem of human resource
management, namely allocating n people into m groups, such that each group satisfies
minimal quality conditions. Each individual is evaluated for every quality by the expert who
grades them using hesitant fuzzy numbers, thus enabling a more realistic mark giving
method by avoiding artificially imposed accuracy. We compare results with crisp models of
Lin and Gen and discuss the significance of fuzzy methodology.
Keywords: hesitant fuzzy sets; multicriteria optimization; possibilistic programming
1
Introduction
Just as in many other areas, project management makes use of mathematical
modeling in cases such as setting strategic goals, formulation of strategies,
selection of human resources and realization of the chosen strategy and control.
Criteria and restrictions of alternatives are also encompassed in the space of
uncertainty and indeterminacy.
Resource Allocation Problem (RAP) is the process of allocating resources among
the various projects or business units for maximization of profit or minimization
of cost [10]. The process of the RAP is focused on finding an optimal allocation of
limited resources to a certain number of tasks while controlling for the given
resource constraint. Resource may be any entity used to accomplish a goal, e.g. a
person, an asset, material, etc. This paper deals with human resource allocation.
We start from a set of individuals who were tested for certain qualities by the
experts. The novelty lies in the fact that marks given by the experts need not to be
exact numbers, but qualitative values, such as: absolutely true, highly true,
moderately true, etc. Furthermore, if experts do have certain hesitations it will be
allowed to choose more than one ‘mark’, thus avoiding to be artificially precise.
Methodology which allows us to model indeterminacy is fuzzy sets theory, which
is particularly well designed for dealing with non-probabilistic uncertainties.
Following the work of Saaty et al. [20] we observe marketing sector vacancies.
– 185 –
Z. Ciric et al.
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
We expand their work in two directions: expanding the grading system and,
indirectly, the coefficients of model constraints which become fuzzy. On the other
hand, we allow each candidate to be placed at the position best suited for his
qualities, which is not necessarily the position he/she would apply for in the first
place.
Chapter two focuses on literature review, both of human resources allocation and
hesitant fuzzy sets. Chapter three presents the data and variables. We pose the
model and perform an in-depth analysis and construction of constraints’
coefficients. LINDO software is used for solving binary variables problems.
Results are presented and differences between results of similar studies are
pointed out. Finally, the last chapter concludes and the guidelines for future
research are given.
2
Literature Review
We begin with the review of papers which inspired us the most, dealing with
human resources allocation. Due to the considerable amount of subjectivity
involved in hiring people, it is helpful to have prioritization techniques to deal
with intangibles and make the process more objective, Saaty [18] and Saaty et al.
[19]. They propose Analytic Hierarchy Process (AHP), and linear programming to
rate and derive the best combination of people assigned to jobs [24] such as
Rezaei et al. [15].
Filho et al. [6] uses the Constraint Satisfaction Problem approach to solve human
resource allocation problems in cooperative health services. They proposed a new
tool for planning human resources utilization in hospital plants, i.e. health units.
They used simulations for measuring the performance of the proposed heuristics.
Another contribution of this work was the mathematical modeling of the unary,
multiple, numeric and implicit constraints.
Ballesteros-Pérez et al. [2] described a new application to key sociometrical
concepts for the selection of workers within organizations in area of project
development, in order to assure the achieving of the highest possible overall work
efficiency from the viewpoint of social interaction.
Silva and Costa [21] focused on software development projects and presented a
methodology based on dynamic programming for assigning human resources to
the observed projects [24]. The methodology takes into account the complexity of
each project and the existing capabilities of staff and the skills required for the
project. A simulation is used to demonstrate the decision model.
The concept of Hesitant Fuzzy Sets (HFS) is a new one, dating back to Torra [23].
HFS are a generalization of intuitionistic fuzzy sets of Atanassov and Gargov [1]
who allowed each fuzzy membership function to have a degree of indeterminacy.
– 186 –
Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
Torra defines a HFS through a function that returns a set of membership values for
each element in the domain. The applications of HFS followed various directions,
mostly in generalization of the existing results dealing with (ordinary) fuzzy sets.
Rodríguez et al. [17] focused on group decision making (GDM) and hesitant
situations that could arise from multiple opinions. They have shown how to
generate comparative linguistic expressions by using a context-free grammar.
Rodríguez et al. [16] extended context-free grammar to that defined in Rodríguez
et al. [17], and further expanded the topics in GDM in terms of expressing and
modeling doubt between different linguistic terms, requiring richer expressions to
express the experts’ knowledge more accurately.
On the other line of reasoning, but also tackling the decision making processes,
Xia and Xu [25] presented an extensive study on hesitant fuzzy information
aggregation techniques and their application in decision making with anonymity.
They have developed certain hesitant fuzzy operational rules based on the
interconnection between the hesitant fuzzy set and the intuitionistic fuzzy set, as
well as a series of aggregation operators for dealing with various situations, and
risk. Portik and Pokoradi have applied the fuzzy rule based risk assessment and
the authors use it in the decision making process [14].
3
Methodology and Model Explanation
In this chapter we shall present in detail the proposed methodology and provide
the readers with the model for human resources allocation to different positions in
marketing sector. When evaluating candidates, HR management is confronted
with a significant number of tradeoffs over a diverse range of criteria.
Multiple-objective optimization problems have gained momentum in researchers’
community with various backgrounds since early 1960. In the realm of
multiobjective optimization problem, multiobjective functions are optimized
simultaneously. Therefore, there does not necessarily exist a solution that is best
with respect to all objectives due to the incommensurability and confliction among
objectives. A solution may be the best in one objective but underperforming in
another. The so-called Pareto optimal solutions are a set of solutions for the
multiple-objective case which cannot simply be compared with each other, i.e. no
improvement is possible in any objective function without sacrificing at least one
of the other objective functions.
Fuzzy mathematical programming (FMP) was developed in order to adequately
solve optimization problems which poses non stochastic indeterminacies.
Inuiguchi [7] classifies FMP in three categories based on the types of
indeterminacies: 1) FMP with ambiguities, also known as flexible programming,
developed by Bellman and Zadeh [3], allows fuzzy preferences for the decision
maker. 2) FMP with vagueness deals with fuzzy coefficients in constraints and
– 187 –
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
Z. Ciric et al.
goals. Dubois and Prade [4] were the first researchers who investigated systems of
linear equations with fuzzy coefficients and suggested possible applications of
fuzzy mathematical programming. Dubois [5] observed fuzzy coefficients as
possibilistic distributions of coefficient values, hence the name possibilistic
programming for the second type of FMP. Following the early work of Dubois,
we shall apply possibility theory in our linear model. Finally, third type, robust
programming combines fuzzy coefficients and ambiguous preferences of the
decision makers. Negoita et al. [12] were the first to formulate such a linear
programming problem.
Managing and modeling of uncertainty by different forms of information, used by
experts to provide their preferences, can be done in various ways, including utility
vectors, fuzzy preference relations, linguistic variables, interval values,
multiplicative preference relations, hesitant fuzzy sets.
Hesitant fuzzy sets are a generalization of ‘ordinary’ fuzzy sets and will be used in
possibilistic programming. For every element x from the domain X, membership
function of a HFS A is given by h(x) = Ax, where Ax is any (finite) subset of [0,1].
Suppose that the experts deciding on candidate’s communication skills should
choose a grade from the set of linguistic descriptions, L = {none, very low, low,
medium, high, very high, absolute}. Of course, if unsure of the exact grade, an
expert has a possibility to opt for two or more grades 1. Grades can be naturally
ordered, and modeled by fuzzy sets, as shown in Figure 1.
x
1
0,5 1 1,5 2 2,5 3 3,5 4 4,5 5 5,5
Figure 1
Choosing more than one grade would result in ‘merging’ the graphs of chosen
grades into one convex fuzzy set. It is done by means of envelope which is a
linguistic interval whose limits are minimal and maximal linguistic term, as shown
in Figure 2.
1
Choosing all options is equivalent to total absence of knowledge.
– 188 –
Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
1
1
3
Figure 2
We observe only symmetrical marks, bounded with linear functions, though
different methods of forming asymmetrical functions exist and can be found in
Inuiguchi et al. [8]. We also refer an interested reader to Klir and Yuan [9] for any
comprehensive and in-depth reading on fuzzy sets and fuzzy logic, such as Nagy
et al. ([11]) and Takacs [22].
3.1
The Model
Let us observe a company ‘Comp’ consisting of S sectors (Manufacturing,
Marketing, R&D, etc.) Each sector consists of up to P positions (e.g. Marketing
sector embodies marketing vice president, marketing assistant, customer service
representative, shipping clerk, etc.), i.e. P = maxPi, where Pi is the number of
positions in sector i, i=1,2,…S. Each position employs up to K employees, where
K being maxKij, and Kij is the number of employees on the position j of the sector
i (e.g. there are two positions for marketing VP, 3 for marketing assistant, 6 for
customer service representative, 2 for shipping clerk, etc.)
There is a total of N candidates, each being tested for C characteristics, regardless
of the position he/she apply for. The program chooses based on the test results the
position one is suited best for, and that doesn’t necessarily mean is the position
one applied for. Characteristics include experience, organizational skills,
dependability, computer knowledge, contacts, formal education, etc. These
characteristics are measured by both numerical values and experts’ opinions in
terms of hesitant fuzzy sets, and their systematization with graphical expressions
is given below:
Experience is divided into three broad intensity groups: high (corresponds to 15+
years of experience), medium (6-15 years), and low (up to 5 years). For each
broad group experts are asked to grade the applicant in terms how satisfactory (in
terms of Figures 1 and 2) his actual achievements are, being in a certain intensity
group. Similar reasoning goes for following characteristics: quality contacts and
leadership, which are also divided into high, medium, and low, with further subdivisions by the experts (Figure 3). Organizational and computer skills, as well as
dependability are solely divided into broad groups, without additional experts’
– 189 –
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
Z. Ciric et al.
opinion. Finally, education is divided into PhD, MSc, BSc, and secondary, while
the experts give opinion on the quality of the school attended, thus allowing for
slight differences within the same educational level.
Low
Medium
High
Figure 3
Let us denote by xijk a binary variable which has a value 1 if the candidate k,
k=1…N is chosen for the position i, i=1…P in sector j, j=1…S, otherwise, xijk has
a value 0. Natural constraints imposed on x’s are following:

P
S
N
i
j
k
xijk  1, for every k  1...N .
(1)
One candidate can be employed at exactly one position.

P
S
N
i
j
k
xijk N
(2)
Total number of candidates exceeds number of positions.
Each sector is choosing candidates in order to maximize it’s manpower:

P
S
N
i
j
k
cijk xijk
(3)
Coefficients cijk in the linear expression (3) are fuzzy numbers of triangular, or
trapezoidal shape and are obtained through summation of grades for each position.
Analytical and graphical expression of a trapezoidal number A = (a, b, α, β) is
given by following membership function and in Figure 4:
 ax
,a   x 
1 


 A ( x)  1
,a  x  b,

xb
1   , b  x  b  

– 190 –
Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
Figure 4
The linear combination of fuzzy trapezoidal numbers, which is present as a
constraint expression as well as a goal function, is again a number of the same
shape. If we denote by (aijk, bijk, αijk, βijk) coefficient cijk, the total manpower TM,
defined at (3) is given by:






 i, j ,k aijk xijk , i, j ,k bijk xijk , i, j ,k  ijk xijk , i, j ,k  ijk xijk    Ax , Bx , x ,  x 


The possibilistic linear function is not uniquely determined, therefore its
maximization, as well as constraints of the form  Ax, Bx, x,  x  M may
not be completely meaningful. For overcoming this problem we introduce a socalled ‘fuzzy inequality’,   A, B  , where A and B are two fuzzy sets. If the
possiblistic distribution µA of a possibilistic variable α, the measurements of
possibility and necessity of an event that α belongs to the set B ([5]):
PossA(B) = supxmin(µA(x), µB(x))
(4)
NesA(B) = infxmax(1 - µA(x), µB(x)),
(5)
where µB is a membership function of a set B. PossA(B) estimates the level up to
which it is possible that possibilistic variable α belongs to the set B. on the other
side, the expression NesA(B) measures the level of necessity up to which is certain
that α belongs to the set B. Let α be a possibilistic variable and B = (-∞, g], a crisp
set. One could easily obtain the following: Pos(α ≤ g) = sup{ µA(x) | x≤ g} and
Nes(α ≤ g) = 1 - sup{ µA(x) | x> g}, which is exhibited in Figure 5. Similarly, for B
= [g, +∞) we obtain following equations presented in Figure 6:
Pos(α ≥ g) = sup{ µA(x) | x≥ g} and Nes(α ≥ g) = 1 - sup{ µA(x) | x < g}.
Figure 5
– 191 –
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
Z. Ciric et al.
Figure 6
The constraints on the number of people in each position are given by:

x
i , j ijk
 pij , where i is the number of sectors, i=1…S and j is the number of
positions in each sector, j=1…P.
Maximization of the total human potential of the company is done by maximizing
the right spread of the possibilistic distribution, allowing for the best possible
marks to be incorporated in the optimal team:
 
Maximize
i , j ,k
ijk

 bijk xijk
(6)
One of the possible restrictions is a funding restriction. Each employee has
suggested his expected and minimal wage, we and wmin respectively. We would
like not to break funds F with high necessity, of e.g. h = 0.8 or h = 0.9. One easily
concludes from Figure 5 that this constraint is equivalent to:
x w  F
 x w  w
i
i
ei
ei
min i

 1 h
(7)
Other restrictions are possible, namely if we want to impose a minimal mark value
in the certain job group, e.g. average computer skills among marketing assistants
should be at least M. Since we observe all the marks as possibilistic distribution
functions, we can pose this requirement differently, namely, we could demand that
the minimal average computer skill is necessarily greater than M, where we use
high necessity measure of e.g. h = 0.8 or h = 0.9, as presented in Figure 6. The set
of restrictions is given by:
a x
a x
M
i i
 1 h
(8)
i i
– 192 –
Acta Polytechnica Hungarica
4
Vol. 12, No. 3, 2015
Results and Discussion
Numerical example is provided in this section. Data from 30 applicants were
collected, and only the marketing sector has been covered. Data can be found in
Appendix 12. In addition, applicants were asked to suggest their salary in terms of
expected and minimum value. Values of salaries are given in Appendix 2. This
group was tested for education levels, contacts, experience, leadership skills,
organizational, computer secretarial, and communicational skills, as well as
dependability. Not all skills are necessary for each position, namely: VP
Marketing (up to two people) are tested for experience, contacts, education and
leadership. Marketing assistants (MA, up to three) for organization, computer
skills and education, Customer service representatives (CSR, up to four) for
education, computer, secretarial and communicational skills. Finally, Shipping
clerks (SC, up to five) were tested for dependability, organizational and computer
skills. Objective function was to maximize the total manpower of future
employees.
Results without limited funds are presented in Table 1. Results include restrictions
imposed on VPs to have an average score of at least 20. For MAs for having an
average mark on education and organization at least 8, on CSRs to have average
communicational skills at least 2.5.
Table 1
Optimal choice of applicants, necessity coefficient 0.8. Unlimited funds
Position
VP
MA
CSR
SC
Ordinal number of the applicant
13, 23
7, 12, 22
4, 9, 26, 30
8, 15, 16, 21, 29
Costs
185-190
285-295
267-280
192-207
Total human capital equals 426.5. Total costs up to 972.
However, if the funds are decreased to 600 (in thousands of US dollars per
annum) we observe the results in Table 2:
Table 2
Optimal choice of applicants, necessity coefficient 0.8.Funds limited to 600
Position
VP
MA
CSR
SC
Ordinal number of the applicant
7, 12
4, 8, 16
6, 11, 15, 25
9, 14, 20, 27, 29
2
Costs
195-200
100-108
138-150
121-138
Applicants are ordered from A.A. through B.D., and in tables in current chapter we used
ordinal notation: A.A. is 1st,...B.D. is 30th.
– 193 –
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
Z. Ciric et al.
Total human capital is decreased to 355.4. Total costs up to 596. Major decrease
in human capital is due to choosing applicants for CSR and SC positions with
lower education levels, as well as low contacts and leadership skills. Even though
these characteristics influence the total score, they are almost irrelevant for the
positions in question, and chosen applicants show excellent results in
characteristics needed for those positions (e.g. high computer and
communicational skills).Further decrease in funding is presented in Table 3a and
3b, together with the change in desired necessity levels, from 0.8 to 0.95.
Table 3a
Further decrease in funds, F = 500, h = 0.8
Position
VP
MA
CSR
SC
Ordinal number of the applicant
7, 12
4, 8, 16
9, 15, 25, 29
2, 14, 18, 24, 27
Costs
195-200
100-108
97-106
76-88
Human capital is slightly decreased to 329.2, while funds are at most 502, though
crossing the limit by 2000 is unlikely, since necessity level is 0.8. For extreme risk
averse investors, Table 3b exhibits the same funds, with necessity level 0.95.
Table 3b
Further decrease in funds, F = 500, h = 0.95
Position
VP
MA
CSR
SC
Ordinal number of the applicant
7, 30
4, 8, 16
9, 15, 25, 29
2, 14, 18, 24, 27
Costs
187-195
100-108
97-106
76-88
HC for this constellation equals 326.5. Maximal amount of money needed is
almost certainly less than 497. Huge savings are possible if focusing on applicants
with lower levels of formal education, yet skillful.
Conclusion
We tried to present a possible application of hesitant fuzzy sets in the field of
human resources allocation. Certain qualities of the applicants cannot be measured
precisely, and an expert’s opinion is needed, which as well may not be precise, or
the expert might have doubts or hesitate to give a final mark. Hesitant fuzzy sets
are designed to perfectly model these situations. We suggested possible objective
functions, as well as an array of possible constraints. There were no restrictions on
number of positions one applicant could apply. We tested the model on 30
applicants to Marketing sector of a company wanting to expand to 14 new
positions. Results show that insignificant loss in total human potential, measured
by sum of marks in all observed characteristics, can lead to substantial savings.
– 194 –
Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
Possible limitation might be the omission of interaction among cooperative
individuals. When choosing a team of people it is relevant to measure a level of
synergy within a team and test whether a team contributes more (or less) than the
sum of independent performances of each team member. Surely it is possible to
employ other mathematical methods in order to model synergy levels. However,
the focus of this paper was on presenting a new method in decision making and
mark giving, but this interesting topic will be fully investigated in our future work.
Appendix 1: List of 30 applicants and their grades in various characteristics
APPLICANTS
A.A. A.B. A.C. A.D. A.E
A.F. A.G. A.H. A.I.
A.J. A.K. A.L. A.M. A.N. A.O. A.P. A.Q. A.R. A.S. A.T. A.U. A.V. A.W. A.X. A.Y. A.Z. B.A. B.B. B.C. B.D.
Experience
2
1,5
2,3
2,0
5,2
5,0
5,4
3
4,4
4,2
3,2
5,2
7,2
2,4
3,1
3,2
2,4
3,2
2,4
4,4
5,3
4,3
6,6
3,2
2,3
4,4
3,2
4,3
3,4
5,5
2
1,4
2,2
2,0
5,1
5,0
5,2
3
4,2
4,1
3,1
5,1
7,1
2,3
3,0
3,1
2,2
3,1
2,2
4,3
5,2
4,2
6,4
3,1
2,2
4,3
3,1
4,2
3,3
5,4
2
1,3
2,1
2,0
5,0
5,0
5,0
3
4
4
3
5,1
7,1
2,2
2,9
3
2
3
2
4,3
5,1
4,2
6,2
3,1
2,1
4,2
3,1
4,2
3,2
5,4
Education
VP Marketing
Contact
Leadership
Computer
Marketing assistant
Organization
Education
Computer
Education
Secretarial
Customer service representative
2
1,2
2
2,0
4,9
5,0
4,8
3
3,8
3,9
2,9
5,0
7,0
2,1
2,8
2,9
1,8
2,9
1,8
4,2
5
4,1
6
3
2
4,1
3
4,1
3,1
5,3
2,5
2,3
3,5
3,5
5,2
5,2
6,2
3,4
4,3
4,8
3,9
5,9
6,8
3,3
3,6
3,8
2,7
3,3
3,0
4,4
5,5
5,3
6,2
3,2
3,3
5,2
3,5
4,7
3,8
5,6
2,4
2,2
3,4
3,4
5,0
5,1
6,0
3,2
4,2
4,6
3,8
5,8
6,7
3,2
3,5
3,7
2,6
3,2
3,0
4,3
5,5
5,2
6,1
3,2
3,2
5,1
3,5
4,6
3,7
5,5
2,4
2,2
3,4
3,4
4,8
5,1
5,8
3,0
4,1
4,4
3,8
5,7
6,6
3,2
3,5
3,6
2,6
3,2
3,0
4,2
5,5
5,2
6,1
3,2
3,2
5,1
3,5
4,5
3,7
5,4
2,3
2,1
3,3
3,3
4,6
5,0
5,6
2,8
4,0
4,2
3,7
5,6
6,5
3,1
3,4
3,5
2,5
3,1
3,0
4,1
5,5
5,1
6,0
3,2
3,1
5,0
3,5
4,4
3,6
5,3
2,1
1,8
4,0
3,7
6,2
5,7
7,2
2,0
3,8
5,8
3,5
6,0
7,5
3,5
2,1
2,0
1,9
1,7
3,3
3,2
5,2
5,5
7,3
1,5
3,0
5,0
3,2
5,0
2,1
4,1
2,0
1,7
4,0
3,6
6,1
5,5
7,1
2,0
3,7
5,6
3,5
6,0
7,5
3,4
2,0
2,0
1,8
1,6
3,2
3,1
5,1
5,5
7,2
1,4
3,0
5,0
3,1
5,0
2,0
4,0
2,0
1,6
4,0
3,6
6,0
5,3
7,1
2,0
3,6
5,4
3,5
6,0
7,5
3,4
1,9
2,0
1,7
1,5
3,2
3,0
5,1
5,5
7,2
1,3
3,0
5,0
3,0
5,0
2,0
4,0
1,9
1,5
4,0
3,5
5,9
5,1
7,0
2,0
3,5
5,2
3,5
6,0
7,5
3,3
1,8
2,0
1,6
1,4
3,1
2,9
5,0
5,5
7,1
1,2
3,0
5,0
2,9
5,0
1,9
3,9
2,8
2,8
4,1
4,3
5,2
5,3
6,5
3,7
4,2
5,1
4,3
6,3
6,6
3,7
3,8
4,2
2,8
3,3
3,2
4,3
5,6
5,7
6,0
3,3
3,8
5,5
3,6
4,8
4,0
5,6
2,7
2,7
4
4,2
5,1
5,2
6,5
3,5
4,1
5
4,2
6,2
6,4
3,5
3,7
4
2,7
3,2
3,1
4,2
5,5
5,5
6 3,22
3,7
5,4
3,4
4,7
4
5,5
2,6
2,6
4
4,1
5,1
5,1
6,5
3,3
4,1
5
4,1
6,2
6,2
3,3
3,7
4
2,6
3,2
3,1
4,1
5,4
5,3
6
3,2
3,6
5,4
3,2
4,7
4
5,5
2,5
2,5
4
4,0
5,0
5,0
6,5
3,1
4
4,9
4
6,1
6,0
3,1
3,6
3,8
2,5
3,1
3
4
5,3
5,1
6
3,1
3,5
5,3
3
4,6
4
5,4
1,3
2
3,1
4,1
1,2
2,3
4,0
3
1,4
2,2
3,1
4,7
2,0
2,3
3,3
4,3
1,2
2,1
1
2,4
3,3
4,4
1,2
2,1
3
4,2
1,5
2,2
3,4
4,2
1,2
2
3
4,0
1,1
2,2
4,0
3
1,2
2,1
3
4,6
2,0
2,2
3,2
4,2
1,1
2
1
2,3
3,2
4,2
1,1
2
3
4,1
1,5
2,1
3,3
4,1
1,1
2
2,9
4,0
1,1
2,2
4,0
3
1
2,1
3
4,6
2,0
2,2
3,1
4,1
1,1
2
1
2,2
3,1
4
1,1
2
3
4,1
1,5
2,1
3,2
4,1
1
2
2,8
3,9
1,0
2,1
4,0
3
0,8
2
2,9
4,5
2,0
2,1
3,0
4
1
1,9
1
2,1
3
3,8
1
1,9
3
4
1,5
2
3,1
4
1,1
2,1
2,9
4,1
0,9
2,1
3,1
5
1,1
2
3,2
4,5
1,0
2,2
3,2
4,2
1,3
2,1
1,2
2,3
3,1
4,4
1,1
2,1
3,2
4
1,4
2,3
3,2
4,1
1
2
2,8
4,0
0,8
2,0
3,0
5
1
2
3,1
4,5
1,0
2,1
3,1
4,1
1,2
2
1,1
2,2
3
4,3
1
2
3,1
4
1,3
2,2
3,1
4
1
2
2,7
4,0
0,8
2,0
3,0
5
1
2
3,1
4,5
1,0
2,1
3,1
4,1
1,1
2
1,1
2,1
3
4,2
1
2
3
4
1,2
2,2
3,1
4
0,9
1,9
2,6
3,9
0,7
1,9
2,9
5
0,9
2
3
4,5
1,0
2,0
3,0
4
1
1,9
1
2
2,9
4,1
0,9
1,9
2,9
4
1,1
2,1
3
3,9
2,1
1,8
4
3,7
6,2
5,7
7,2
2
3,8
5,8
3,5
6,0
7,5
3,5
2,1
2
1,9
1,7
3,3
3,2
5,2
5,5
7,3
1,5
3
5
3,2
5
2,1
4,1
2
1,7
4
3,6
6,1
5,5
7,1
2
3,7
5,6
3,5
6
7,5
3,4
2
2
1,8
1,6
3,2
3,1
5,1
5,5
7,2
1,4
3
5
3,1
5
2
4
2
1,6
4
3,6
6
5,3
7,1
2
3,6
5,4
3,5
6
7,5
3,4
1,9
2
1,7
1,5
3,2
3
5,1
5,5
7,2
1,3
3
5
3
5
2
4
1,9
1,5
4
3,5
5,9
5,1
7
2
3,5
5,2
3,5
6
7,5
3,3
1,8
2
1,6
1,4
3,1
2,9
5
5,5
7,1
1,2
3
5
2,9
5
1,9
3,9
1,1
2,1
2,9
4,1
0,9
2,1
3,1
5
1,1
2
3,2
4,5
1
2,2
3,2
4,2
1,3
2,1
1,2
2,3
3,1
4,4
1,1
2,1
3,2
4
1,4
2,3
3,2
4,1
1
2
2,8
4
0,8
2
3
5
1
2
3,1
4,5
1
2,1
3,1
4,1
1,2
2
1,1
2,2
3
4,3
1
2
3,1
4
1,3
2,2
3,1
1
2
2,7
4
0,8
2
3
5
1
2
3,1
4,5
1
2,1
3,1
4,1
1,1
2
1,1
2,1
3
4,2
1
2
3
4
1,2
2,2
3,1
4
0,9
1,9
2,6
3,9
0,7
1,9
2,9
5
0,9
2
3
4,5
1
2
3
4
1
1,9
1
2
2,9
4,1
0,9
1,9
2,9
4
1,1
2,1
3
3,9
2,1
1,8
4
3,7
6,2
5,7
7,2
2
3,8
5,8
3,5
6
7,5
3,5
2,1
2
1,9
1,7
3,3
3,2
5,2
5,5
7,3
1,5
3
5
3,2
5
2,1
4,1
2
1,7
4
3,6
6,1
5,5
7,1
2
3,7
5,6
3,5
6
7,5
3,4
2
2
1,8
1,6
3,2
3,1
5,1
5,5
7,2
1,4
3
5
3,1
5
2
4
4
Communicatio Dependability
Organization
Shipping clerk
Computer
2
1,6
4
3,6
6
5,3
7,1
2
3,6
5,4
3,5
6
7,5
3,4
1,9
2
1,7
1,5
3,2
3
5,1
5,5
7,2
1,3
3
5
3
5
2
4
1,9
1,5
4
3,5
5,9
5,1
7
2
3,5
5,2
3,5
6
7,5
3,3
1,8
2
1,6
1,4
3,1
2,9
5
5,5
7,1
1,2
3
5
2,9
5
1,9
3,9
1,5
2,0
3,3
4,0
2,8
3,4
4,8
3,3
6,3
3,3
3,3
5,1
3,5
2,7
2,9
3,5
1,5
2,0
1,8
2,6
3,9
4,8
3,2
1,9
3,1
4,4
2,0
3,2
2,9
4,1
1,4
2,0
3,2
4,0
2,6
3,3
4,7
3,2
6,2
3,2
3,2
5,0
3,4
2,6
2,8
3,5
1,4
2,0
1,7
2,5
3,8
4,7
3,1
1,8
3,0
4,3
2,0
3,1
2,8
4,0
1,4
2,0
3,2
4,0
2,4
3,2
4,5
3,2
6,1
3,1
3,1
5,0
3,2
2,6
2,7
3,5
1,4
2,0
1,6
2,5
3,7
4,7
3,1
1,8
3,0
4,2
2,0
3,1
2,7
4,0
1,3
2,0
3,1
4,0
2,2
3,1
4,4
3,1
6,0
3,0
3,0
4,9
3,1
2,5
2,6
3,5
1,3
2,0
1,5
2,4
3,6
4,6
3,0
1,7
2,9
4,1
2,0
3,0
2,6
3,9
1,3
2,0
2,9
3,8
1,7
2,7
3,8
4,0
6,0
2,4
3,2
4,7
2,4
2,3
3,2
4,1
1,5
2,3
1,3
2,7
3,5
4,4
2,1
2,3
3,0
4,1
1,7
2,6
3,3
4,4
1,2
2
2,8
3,7
1,6
2,6
3,7
4
6
2,3
3,1
4,6
2,3
2,2
3,1
4
1,4
2,2
1,2
2,6
3,4
4,3
2
2,2
3
4
1,6
2,5
3,2
4,3
1,2
2
2,7
3,6
1,5
2,4
3,6
4
6
2,3
3,1
4,5
2,2
2,1
3
4
1,4
2,1
1,2
2,4
3,3
4,2
2
2,1
3
4
1,4
2,5
3,2
4,1
1,1
2
2,6
3,5
1,4
2,3
3,5
4
6
2,2
3
4,4
2,1
2
2,9
3,9
1,3
2
1,1
2,3
3,2
4,1
1,9
2
3
3,9
1,3
2,4
3,1
4
1,55
1,8
2,6 3,05 3,05 3,55 4,25
4 2,75
3,1
3,2 4,85
4,1
1,5
1,7
2,5
1,5
1,6
2,4
1,4
1,5
2,3
1,3
2
3,1
1,2
2
1,1
2
1
2
1,1
1
3
2,3 3,15
3,7 1,85 2,65
1,8 3,35
4,2 4,35 3,85 2,65 2,75
4,2
2,3
3,3
3,3
4,8
4
2,2
3,1
3,6
1,8
2,6
1,7
3,3
4,1
4,3
3,8
2,6
2,7
4,1
2,2
3,2
3,2
4,7
4,6
4
2,1
3,1
3,5
1,6
2,5
1,5
3,3
4
4,2
3,8
2,6
2,5
4
2,1
3,1
3,1
4,5
4,5
3,9
2
3
3,4
1,5
2,4
1,4
3,2
3,9
4,1
3,7
2,5
2,4
3,9
2
3
3
4,4
3,1
4,7
2
2,3
3,3
4,3
1,2
2,1
1
2,4
3,3
4,4
1,2
2,1
3
4,2
1,5
2,2
3,4
4,2
2,1
3
4,6
2
2,2
3,2
4,2
1,1
2
1
2,3
3,2
4,2
1,1
2
3
4,1
1,5
2,1
3,3
4,1
2,1
3
4,6
2
2,2
3,1
4,1
1,1
2
1
2,2
3,1
4
1,1
2
3
4,1
1,5
2,1
3,2
4,1
0,8
2
2,9
4,5
2
2,1
3
4
1
1,9
1
2,1
3
3,8
1
1,9
3
4
1,5
2
3,1
4
1,1
2
3,2
4,5
1
2,2
3,2
4,2
1,3
2,1
1,2
2,3
3,1
4,4
1,1
2,1
3,2
4
1,4
2,3
3,2
4,1
2
3
3,5
4,2
4
2,7
3
3
3,4
4,2
4
2,6
2,9
2,9
3,3
4,1
4
2,5
4,1
1,2
2,3
4
3
14
3
4
1,1
2,2
4
3
2,9
4
1,1
2,2
4
3
2,8
3,9
1
2,1
4
2,1
2,9
4,1
0,9
2,1
2
3
3,1
4,8
3
3
2,9
2,9
2,2
1,2
1
3
3,1
5
2
2,8
4
0,8
3
5
1
2
3,1
4,5
1
2,1
3,1
4,1
1,2
2
1,1
2,2
3
4,3
3,1
4
1,3
2,2
3,1
1
2
2,7
4
0,8
2
3
5
1
2
3,1
4,5
1
2,1
3,1
4,1
1,1
2
1,1
2,1
3
4,2
1
2
3
4
1,2
2,2
3,1
4
0,9
1,9
2,6
3,9
0,7
1,9
2,9
5
0,9
2
3
4,5
1
2
3
4
1
1,9
1
2
2,9
4,1
0,9
1,9
2,9
4
1,1
2,1
3
3,9
– 195 –
1
4
Multicriteria HR Allocation Based on Hesitant Fuzzy Sets and Possibilistic Programming
Z. Ciric et al.
Appendix 2: Expected and minimal salaries per annum (in thousands USD), per applicant
applicant
1
2
3
4
5
7
8
9
10
11
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
exp
15
15
41
55
60
65 100
6
20
40
60
42 100 100
18
21
43
15
15
17
40 100
95
90
16
22
90
17
55
23
95
min
13
13
38
53
55
60
18
36
54
38 100
16
20
39
13
14
15
35
90
88
13
20
86
15
52
19
92
95
12
97
96
References
[1]
Atanassov, K.; Gargov, G: Interval Valued Intuitionistic Fuzzy Sets, Fuzzy
Sets and Systems, Vol. 3, No. 3, 1989, pp. 343-349
[2]
Ballesteros-Pérez, P.; González-Cruz, C.; Fernández-Diego, M: Human
Resource Allocation Management in Multiple Projects Using Sociometric
Techniques, International Journal of Project Management, Vol. 30, No. 8,
2012, pp. 901-913
[3]
Bellman, R. E.; Zadeh, L. A: Decision-Making in a Fuzzy Environment.
Management Sci. Vol. 17, 1970, pp. 141-164
[4]
Dubois, D.; Prade, H: Systems of Linear Fuzzy Constraints, Fuzzy Sets and
Systems, Vol. 3, 1980, pp. 37-48
[5]
Dubois, D: Linear Programming with Fuzzy Data, in: J. C. Bezdek (Ed.),
Analysis of Fuzzy Information 3: Applications in Engineering and Science,
CRC Press, Boca Raton, FL, 1987, pp. 241-263
[6]
Filho, C. F. F. C.; Rocha, D. A. R.; Costa, M. G. F.; Pereira, W. C. A:
Using Constraint Satisfaction Problem Approach to Solve Human Resource
Allocation Problems in Cooperative Health Services, Expert Systems with
Applications, Vol. 39, No. 1, 2012, pp. 385-394
[7]
Inuiguchi, M.; Sakawa, M.; Kume, Y: The Usefulness of Possibilistic
Programming in Production Planning Problems, Int. J. Prod. Economics,
Vol. 33, 1994, pp. 45-52
[8]
Inuiguchi, M.; Tanino, T.; Sakawa, M: Membership Function Elicitation in
Possibilistic Programming Problems, Fuzzy Sets and Systems, Vol. 111,
No. 1, 2000, pp. 29-45
[9]
Klir, G.; Yuan: Fuzzy Sets and Fuzzy Logic: Theory and Applications,
Prentice Hall PTR, 2005
[10]
Lin, C-M; Gen, M: Multi-Criteria Human Resource Allocation for
Solvingmultistage
Combinatorial Optimization Problems
Using
Multiobjective Hybrid Genetic Algorithm, Expert Systems with
Applications, Vol. 34, 2008, pp. 2480-2490
[11]
Nagy, K.; Diveki, Sz.; Odry, P.; Sokola, M.; Vujicic, M: A Stochastic
Approach to Fuzzy Control, Acta Polytechnica Hungarica, Journal of
Applied Sciences, Vol. 9, No. 6, 2012, pp. 29-48
[12]
Negoita, C. V.; Minoiu, S.; Stan, E: On Considering Imprecision in
Dynamic Linear Programming. Economic Comput. Economic Cybernet.
Stud. Res, Vol. 3, 1976, pp. 83-95
– 196 –
Acta Polytechnica Hungarica
Vol. 12, No. 3, 2015
[13]
Osman, M. S.; Abo-Sinna, M. A.; Mousa, A. A: An Effectivegenetic
Algorithm Approach to Multiobjective Resource Allocationproblems,
Applied Mathematics and Computation, Vol. 163, 2005, pp. 755-768
[14]
Portik, T.; Pokorádi, L: The Summarized Weighted Mean of Maxima
Defuzzification and Its Application at the End of the Risk Assessment
Process, Acta Polytechnica Hungarica, Vol. 11, No. 3, 2014, pp. 167-180
[15]
Rezaei, P.; Rezaie, K.; Nazari-Shirkouhi, S.; Jamalizadeh Tajabadi, M. R:
Application of Fuzzy Multi-Criteria Decision Making Analysis for
Evaluating and Selecting the Best Location for Construction of
Underground Dam, Acta Polytechnica Hungarica, Vol. 10, No. 7, 2013, pp.
187-205
[16]
Rodríguez, R. M.; Martı́nez, L.; Herrera, F: A Group Decision Making
Model Dealing with Comparative Linguistic Expressions Based on Hesitant
Fuzzy Linguistic Term Sets, Information Sciences, Vol. 241, 2013, pp. 2842
[17]
Rodríguez, R. M.; Martı́nez, L.; Herrera, F: Hesitant Fuzzy Linguistic Term
Sets for Decision Making, IEEE Transactions on Fuzzy Systems, Vol. 20,
No. 1, 2012, pp. 109-119
[18]
Saaty, T. L: Decision Making with Dependence and Feedback: the Analytic
Hierarchy Process, RWS Publications, 4922 Ellsworth Avenue, Pittsburgh
PA 15213, 1996
[19]
Saaty, T. L.; Vargas, L. G.; Dellmann, K: The Allocation of Intangible
Resources: the Analytical Hierarchy Process and Linear Programming,
Socio-Economic Planning Sciences, Vol. 37, 2003, pp. 169-184
[20]
Saaty, T. L.; Peniwatib, K.; Shanga, J. S: The Analytic Hierarchy Process
and Human Resource Allocation: Halfthe Story, Mathematical and
Computer Modelling, Vol. 46, 2007, pp. 1041-1053
[21]
Silva, L. C.; Costa, A. P. C. S: Decision Model for Allocating Human
Resources in Information System Projects, International Journal of Project
Management, Vol. 31, No. 1, 2013, pp. 100-108
[22]
Takács, M: Multilevel Fuzzy Approach to the Risk and Disaster
Management, Acta Polytechnica Hungarica, Journal of Applied Sciences,
Vol. 7, No. 4, 2010, pp. 91-102
[23]
Torra, V: Hesitant Fuzzy Sets, International Journal of Intelligent Systems,
Vol. 25, 2010, pp. 529-539
[24]
Zemova, B.; Talasova, J: Fuzzy Sets in HR Management, Acta
Polytechnica Hungarica, Journal of Applied Sciences, Vol. 8, No. 3, 2011,
pp. 113-124
[25]
Xia, M.; Xu, Z: Hesitant Fuzzy Information Aggregation in Decision
Making, International Journal of Approximate Reasoning, Vol. 52, 2011,
pp. 395-407
– 197 –